Predicting stress, strain and deformation fields in materials and structures with graph neural networks

Developing accurate yet fast computational tools to simulate complex physical phenomena is a long-standing problem. Recent advances in machine learning have revolutionized the way simulations are approached, shifting from a purely physics- to AI-based paradigm. Although impressive achievements have been reached, efficiently predicting complex physical phenomena in materials and structures remains a challenge. Here, we present an AI-based general framework, implemented through graph neural networks, able to learn complex mechanical behavior of materials from a few hundreds data. Harnessing the natural mesh-to-graph mapping, our deep learning model predicts deformation, stress, and strain fields in various material systems, like fiber and stratified composites, and lattice metamaterials. The model can capture complex nonlinear phenomena, from plasticity to buckling instability, seemingly learning physical relationships between the predicted physical fields. Owing to its flexibility, this graph-based framework aims at connecting materials’ microstructure, base materials’ properties, and boundary conditions to a physical response, opening new avenues towards graph-AI-based surrogate modeling.


Details of the FE modeling
Periodic boundary conditions (PBCs) To analyze the influence of micro-and meso-structure of material systems on their macroscopic mechanical behavior, the application of periodic boundary conditions (PBCs) on representative volume elements (RVEs) is needed. In this work, unidirectional fiber composite and stratified composite RVEs are subject to a 2D macroscopic deformation gradient under PBCs by enforcing: where and are displacements of pair of points periodically located on two opposite boundaries of the RVE, and and are the corresponding initial coordinates; is the identity tensor. Using two virtual nodes, associated to the top and bottom, and left and right edges, respectively, the following macroscopic deformation is prescribed: ], corresponding to uniaxial tension or compression conditions (depending on the sign of 11 ), in which 11 is the uniaxial component of the Green-Lagrangian strain tensor, , used as a measure for finite strains. For more details on PBCs, please refer to 1 .

Analytical model of wrinkling of interfacial layers
To demonstrate the ability of our model to predict complex mechanics, wrinkling of interfacial layers in stratified composites is considered. A hard interfacial layer is embedded in a compliant matrix subject to plain strain uniaxial compression. Here we report the dilute case i.e., "the matrix stress fields emanating from neighboring layers do not interact, hence the shear at the interface is negligible and interfacial layers behave independently of one another." ref. 56 (main text); the long-wave instability is thus not here considered. Solving the governing equations for the critical instability conditions leads to the following expressions for the critical strain, , and wavelength, (ref. 56 main text): , where and are coefficients that for plain strain read: For strains higher than the critical one, the post-buckling wavelength, , and amplitude, , can be obtained as (ref. 56 main text): The plots in Fig. S7C-D are derived using these equations.

Details of the datasets
Lattice structures generation To generate random finite-size lattice metamaterials, we adopt a bottom-up technique featured in our previous work (ref. 57 main text). With reference to Fig. S1, a squared building block of size and beams' thickness t, and its 90 ∘ -rotated version are combined into a 4 × 4 assembly i.e., the unit cell. A binary matrix is used to represent the unit cell, where 1 and 0 indicate either the first or the second building block. Random binary matrices thus correspond to random unit cells. In this work, we generate 762 different unit cells. Finally, to populate the dataset the resulting architectures are further tessellated to obtain 2 × 2 finite-size structures using = 2.5 and a thicker frame surrounding the structures.

Details of the ML model set-up GNN architecture
Our model consists of an encoder, a message-passing module, and a decoder ( Fig. 1 main text). Let = ( , ℰ) be a computational graph, where represents a set of nodes connected to each other through edges (ℰ). The -th node (in ) brings features in the vector (such as nodal coordinates, and base material properties); similarly, the edge (in ℰ) connecting the -th and -th node has a -dimensional feature vector (such as distance between nodes). The node and edge features, and , are encoded into a larger latent space in the encoder module using the two neural networks, and , respectively. The decoder is characterized by the neural network, , which transforms the latent node features to the output fields. Main core of the model is the message-passing module, which exploits the expressive power of GNNs. The message-passing phase runs for from 1 to steps. Defining the node state, , as the transformed latent node features after message steps, messages are passed and aggregated at node using the information from the neighboring nodes, ∈ ( ) = { ∈ | ( , ) ∈ ℰ}, and corresponding edges, ( , ), as: where is a neural network. The node state is then updated through the neural network : Training of the model is supervised on nodal output fields by learning the parametrized differentiable functions , , , , and (i.e., neural networks) through minimization of a mean absolute error loss function. For prediction of multiple loading steps (i.e., evolution of physical fields), to make the approach more general, we insert two gated recurrent units (GRUs) 2 after the message-passing module. The hidden states of the GRUs are collected and considered as the latent node state , then transformed by the decoder into output fields.  Fig. 4C (main text). Surprisingly, the model predicts a stress field resembling that exhibited before buckling instability occurs (see Fig. 4C main text). This leads to the observation that a physical relationship between global displacement (compression) and stress field is learned by the model.   Movie S1. Deformation and stress field evolution in a unidirectional fiber composite system under displacement boundary conditions. Comparison between FE simulations (on the left) and ML predictions (on the right) of a single RVE subject to tensile displacement along the horizontal direction for five loading steps, from 1 to 8 % of effective applied strain. The ML model is trained and tested on the same five loading steps.
Movie S2. Deformation and stress field evolution in a unidirectional fiber composite system under displacement boundary conditions. Comparison between FE simulations (on the left) and ML predictions (on the right) of a single RVE subject to tensile displacement along the horizontal direction for ten loading steps, from 1 to 8 % of effective applied strain. The ML model is trained on five loading steps and tested on ten steps (five from training).